%I #36 Sep 08 2022 08:45:11

%S 0,1,16,2048,1638400,7247757312,164995463643136,18446744073709551616,

%T 9803356117276277820358656,24178516392292583494123520000000,

%U 271732164163901599116133024293512544256

%N a(0)=0; for n > 0, a(n) = (n+1)^(n-2)*2^(n^2).

%C Discriminant of Chebyshev polynomial U_n (x) of second kind.

%C Chebyshev second kind polynomials are defined by U(0)=0, U(1)=1 and U(n) = 2xU(n-1) - U(n-2) for n > 1.

%C The absolute value of the discriminant of Pell polynomials is a(n-1).

%C Pell polynomials are defined by P(0)=0, P(1)=1 and P(n) = 2x P(n-1) + P(n-2) if n > 1. - _Rigoberto Florez_, Sep 01 2018

%D Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990; p. 219, 5.1.2.

%H Rigoberto Flórez, Robinson Higuita, and Alexander Ramírez, <a href="https://arxiv.org/abs/1808.01264">The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials</a>, arXiv:1808.01264 [math.NT], 2018.

%H Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Florez2/florez8.html">Star of David and other patterns in the Hosoya-like polynomials triangles</a>, Journal of Integer Sequences, Vol. 21 (2018), Article 18.4.6.

%H R. Flórez, N. McAnally, and A. Mukherjees, <a href="http://math.colgate.edu/~integers/s18b2/s18b2.Abstract.html">Identities for the generalized Fibonacci polynomial</a>, Integers, 18B (2018), Paper No. A2.

%H R. Flórez, R. Higuita and A. Mukherjees, <a href="http://math.colgate.edu/~integers/s14/s14.Abstract.html">Characterization of the strong divisibility property for generalized Fibonacci polynomials</a>, Integers, 18 (2018), Paper No. A14.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Discriminant.html">Discriminant</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html">Chebyshev Polynomial of the Second Kind</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PellPolynomial.html">Pell Polynomial</a>

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.

%F a(n) = ((n+1)^(n-2))*2^(n^2), n >= 1, a(0):=0.

%F a(n) = ((2^(2*(n-1)))*Det(Vn(xn[1],...,xn[n])))^2, n >= 1, with the determinant of the Vandermonde matrix Vn with elements (Vn)i,j:= xn[i]^j, i=1..n, j=0..n-1 and xn[i]:=cos(Pi*i/(n+1)), i=1..n, are the zeros of the Chebyshev U(n,x) polynomials.

%F a(n) = ((-1)^(n*(n-1)/2))*(2^(n*(n-2)))*Product_{i=1..n}((d/dx)U(n,x)|_{x=xn[i]}), n >= 1, with the zeros xn[i], i=1..n, given above.

%t Join[{0},Table[(n+1)^(n-2) 2^n^2,{n,10}]] (* _Harvey P. Dale_, May 01 2015 *)

%o (PARI) a(n)=if(n<1,0,(n+1)^(n-2)*2^(n^2))

%o (PARI) a(n)=if(n<1,0,n++; poldisc(poltchebi(n)'/n))

%o (Magma) [0] cat [(n+1)^(n-2)*2^(n^2): n in [1..10]]; // _G. C. Greubel_, Nov 11 2018

%Y Cf. A007701, A127670 (discriminant for S-polynomials), A006645, A001629, A001871, A006645, A007701, A045618, A045925, A093967, A193678, A317404, A317405, A317408, A317451, A318184, A318197.

%K nonn,easy

%O 0,3

%A Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 05 2003

%E Formula and more terms from _Vladeta Jovovic_, Aug 07 2003